Exponential and sub-exponential stability times for the derivative wave equation

Yingte Sun; Siming Li; Xiaoqing Wu

doi:10.3934/dcds.2023007

Article Contents

2023, Volume 43, Issue 6: 2186-2212. Doi: 10.3934/dcds.2023007

This issue Previous Article Asymptotic stability of solitary waves for the

$ 1d $

NLS with an attractive delta potential Next Article Topologies of continuity for Carathéodory parabolic PDEs from a dynamical perspective

Exponential and sub-exponential stability times for the derivative wave equation

1.
School of Mathematical Sciences, Yangzhou University, Yangzhou, Jiangsu 225002, China
2.
School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China

^*Corresponding author: Siming Li
^*Corresponding author: Siming Li

Received: September 2022

Revised: November 2022

Early access: February 7, 2023

Published online: February 7, 2023

The first author is supported by the China Postdoctoral Foundation (No.2021M692717), Jiangsu Province Postdoctoral Foundation (No.2021K163B) and the National Natural Science Foundation of China (No.12101542).

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Abstract

We prove a Nekhoroshev type theorem for the derivative wave equation

$ \begin{equation*} u_{tt} = u_{xx}-mu-(D^{\alpha}u)^3, \quad \ D = \sqrt{-\partial_{xx}+m} \end{equation*} $

under Dirichlet boundary conditions and $ 0\leq \alpha\leq 1 $. More precisely, we prove the sub-exponential long time stability result in a smooth function space and the exponential long time stability result in a modified Sobolev space around the origin for the above equation, where using Birkhoff normal form technique for infinite dimensional Hamiltonian systems and the so-called $ {tame} $ property of the non-linearity. This result is inspired by Bambusi-Grébert (2006, Duke) and a recent work by Feola-Massetti (arXiv: 2207.09986v1).

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Mathematics Subject Classification: Primary: 37K55, 37J40; Secondary: 35B35, 35Q35.

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